GROUP OF ORDER 64 # 61

GROUP # 61

The MAGMA library number for G is 22

GrpPC : G of order 64 = 2^6 PC-Relations: G.1^2 = G.3, G.2^2 = G.4, G.3^2 = G.6, G.4^2 = G.6, G.5^2 = G.6, G.2^G.1 = G.2 * G.5, G.5^G.1 = G.5 * G.6, G.5^G.2 = G.5 * G.6

The center of G is abelian of type [ 2, 4 ]
The orders of the terms of the lower central series are [ 64, 4, 2, 1 ]
The orders of the terms of the upper central series are [ 1, 8, 16, 64 ]
The order of the Frattini subgroup is 16
The exponent of G is 8
G has a unique maximal elementary abelian subgroup which is normal and has order 8. Its centralizer has order 32

The cohomology ring of G is the quotient of a polynomial ring in the variables [ z, y, x, w, v, u, t, s, r, q ] in degrees [ 1, 1, 2, 2, 2, 2, 3, 3, 4, 4 ] by the ideal generated by the relations
[ z^2, z*y, y^2, z*x, y*x + z*v, y*v, z*u + y*u, x^2, x*v, v^2, x*u + y*t, v*u + y*s, z*t + y*t, z*s + y*s, y*w*u + v*s, y*u^2 + x*s + y*r, x*t + x*s + v*s + y*r, v*t + x*s, z*r + y*r, w*u^2 + s^2, u^3 + t*s + u*r, y*w*t + y*u*s + v*r, y*u*t + y*w*s + y*u*s + x*r, t^2 + t*s + s^2 + u*r, w*u*t + u^2*s + z*v*q + s*r, u^2*t + w*u*s + u^2*s + z*v*q + t*r, u*t*s + w*s^2 + u*s^2 + u^2*r + r^2 ]

The Hilbert series for the cohomology ring is -1/(t^4 - 2*t^3 + 2*t - 1)
Its denominator factors as ( t - 1 )^3 ( t + 1 )^1

The Krull dimension of the cohomology ring is 3
The longest regular sequence consists of the generators [ w, q ]
A homogeneous set of parameters is the set [ w, q, u ] of degrees [ 2, 4, 2 ]

The hypercohomolgy spectral sequence has E2 term:

ROW (0) [ y, z ] [ x, v ] [ t, z*v, s ] [ r ] [ y*r ]
ROW (1) [ z + y ] [] [ z*v ]
The spectral sequence satisfies Poincaré duality

Restriction to the Maximal Elementary Abelian Subgroup

Generated by [ G.3 * G.4 * G.6, G.6, G.4 * G.5 * G.6 ] The cohomology ring of E is a polynomial ring in the variables [ z, y, x ]

The images of the generators of the cohomology of G restricted to E are
[ 0, 0, 0, x^2, 0, z^2, z^3 + z^2*x, z^2*x, z^4 + z^3*x + z^2*x^2, z^2*y^2 + y^4 ] in the cohomology of E

The kernel of the restriction to E (Minimal Prime) of the cohomology of G is generated by
[ z, y, x, v ]

This ideal is also the nilradical of the cohomology of G.

Restrictions to Maximal Subgroups

Maximal Subgroup H # 1

The group H is abelian of type [ 8, 2, 2 ]

The cohomology ring of H is a the quotient of a polynomial ring in the variables [ z, y, x, w ] in degrees [ 1, 1, 1, 2 ] by the ideal of relations
[ z^2 ]

The images of the generators of the cohomology of G restricted to H are
[ z, z, z*x, y^2, z*y, y^2 + x^2, y^2*x + x^3 + z*w, z*y^2 + y^3 + z*x^2 + y*x^2 + z*w, y^4 + z*y^2*x + y^3*x + z*x^3 + y*x^3 + x^4 + z*y*w + z*x*w, y^2*w + x^2*w + w^2 ] in the cohomology of H

The kernel of the restriction to H of the cohomology of G is generated by
[ z + y ]

Maximal Subgroup H # 2

The Group H is Isomorphic to the Group of Order 32 Number 13 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.3 * $.5, $.2^2 = $.3, $.3^2 = $.5, $.4^2 = $.5, $.2^$.1 = $.2 * $.5, $.4^$.1 = $.4 * $.5, $.4^$.2 = $.4 * $.5 Generated by [ G.2 * G.3 * G.5 * G.6, G.5, G.2 ]

of type Cyclic(2) x Group(16)#11

The images of the generators of the cohomology of G restricted to H are
[ 0, z + y, z*y + y^2, y^2, z*y + y^2 + z*x + y*x, z*y + z*x + y*x + x^2, z^2*y + z*y^2 + z*y*x + x^3, z*y*x + x^3 + w, z^3*y + z^2*x^2 + x*w, z^3*x + z^2*y*x + z*y*x^2 + v ] in the cohomology of H

The kernel of the restriction to H of the cohomology of G is generated by
[ z ]

Maximal Subgroup H # 3

The Group H is Isomorphic to the Group of Order 32 Number 13 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.3 * $.5, $.2^2 = $.3, $.3^2 = $.5, $.4^2 = $.5, $.2^$.1 = $.2 * $.5, $.4^$.1 = $.4 * $.5, $.4^$.2 = $.4 * $.5 Generated by [ G.1, G.1 * G.3 * G.5 * G.6, G.3 * G.4 * G.5 * G.6 ]

of type Cyclic(2) x Group(16)#11

The images of the generators of the cohomology of G restricted to H are
[ z + y, 0, z*y + y^2 + z*x + y*x, z*y + z*x + y*x, z*x + y*x, z*y + z*x + y*x + x^2, z^2*y + z^2*x + z*y*x + y*x^2 + w, z*y^2 + z^2*x + z*y*x + y*x^2, z^3*x + z^2*y*x + z^2*x^2 + x^4 + y*w, z^3*x + z^2*y*x + z*y*x^2 + v ] in the cohomology of H

The kernel of the restriction to H of the cohomology of G is generated by
[ y ]

The essential cohomology of G is generated as an ideal by
[ z*v ]

The annihilator of the Essential Cohomology has dimension 2 The essential cohomology is a free module over the polynomial subring Q of the cohomology ring of G generated by [ w, q ] .
The essential cohomology is generated as a module over Q by the elements [] [] [ z*v ]

CHECKS

paramflag = true
qregflagflag = true
cmflag = false
essflag = true
centflag = true
bigflag = true
restrictflag = true

THE COMPUTATION IS COMPLETE